'What's to Notice?' printed from http://nrich.maths.org/

The theme for
this month is about 'actions on objects'. All of the activities
we have devised involve some kind of noticing, and several of them
involve new interactivities which we think promote lots of
opportunities to notice. John Mason, our guest editor this
month, has written a lot about the 'discipline of noticing' and how
important it is to the understanding of the deeper structures of
mathematics. So what does this mean and how can we help our
students get better at it - how can we 'educate awareness'?

Rather than think abstractly about this, let's consider what
it looks like in some of this month's offerings.

The Add and
Take-away Path can be construed as an exercise in adding
up and taking away. One might also think of it as a very
early introduction to vectors. But the task has been devised so
that the problem solver is almost forced to notice something.
Perhaps two totals being the same might be a coincidence, but if
the children are encouraged to try some others they notice that so
long as they fix the start and end of a path, the total will always
be the same. To adults that may be obvious but to small children it
is far from being so and they can be quite surprised.
The teacher's role here is in offering a structured task which
has lots of flexibility so that the children can choose any
routes they like. And then the big question - what do they notice
and why do they think this happened?

I really like Area and
Perimeter. Children so often get these muddled up, usually
because they meet them at different times in their school career,
so setting a task where they intentionally have
to attend to both simultaneously is a good way of confronting
the confusion. The usual questions students are asked are about
knowing a formula and calculating the value of either
area or perimeter but here the questions are of a different nature.
They force the child to look at the relationship between the shape,
its area and its perimeter. Long thin shapes may have a small area
and a large perimeter for example; shapes with bits cut out may
have a very small area and a very big perimeter - so the
structure and property of the shape itself become the
important ideas.

Missing Multipliers and
the variants can be used as consolidation games. In each
case the students have to notice the commonalities, framed in the
multiplication operation, so that they can deduce what whole
number factors go where. Similar noticings have to take place
in Finding Factors.
Because many of the questions differ only slightly, asking
which piece of the answer gives the most information demands
noticing, which leads to a clearer understanding of the structure
of the question.
Whilst I was focused on this, two activities from previous months
came to mind.

Tilted
squares is a task we published last month. The team have been
using it a lot with students recently (you can see videos of this
here), and
have refined their thinking about how it might be used, and the
sorts of questions to ask. The students are invited to perform a
certain action (tilting a square) and asked what they notice. The
teacher's role is in helping them to attend to similarities
and differences - and in this case the way that these are
recorded by the teacher on the board helps the students to notice
in a way that they may not have done if they had recorded in other
ways.

One of my favourite tools on the NRICH site is Shuffles.
You'll need to read the instructions first, but as an interactivity
to model actions, and successive actions, on objects, it's an
engaging option. Lots of deciding what to attend to and how to
record it.

In all of these tasks - and the others this month - 'actions
on objects' change something. Learners get caught up in the
detective work of trying to fathom what it is that makes a
difference, and sometimes whether the order of actions matters. If
you're short of something to do and want to refine your own
noticing skill, try 18-hole
Light Golf . It could become addictive.